■ id d ac y
RESEARCH REPORTS DIVISION
NAVAL POSTGRADUATE SCHOOL
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bort Number NPS-61-83-003
IIBRARY
f e IS DIVISION
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MONTEREY, CALIFORNIA 9394Q
NAVAL POSTGRADUATE SCHOOL
Monterey, California
CERENKOV RADIATION FROM
BUNCHED ELECTRON
BEAMS
F.
R.
Buskirk and J.R.
Neighbours
October 19 82
Revised April
1983
Technical
Report
Approved for public releaser distribution unlimited
repared for:
FEDDOCS hief of Naval Research
D 208.14/2:NPS-61 -83-003 rlington, Virginia 22217
NAVAL POSTGRADUATE SCHOOL
Monterey, California
Rear Admiral John Ekelund David Schrady
Superintendent Provost
The work reported herein was supported by the Chief
of Naval Research.
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Cerenkov Radiation from Bunched
Electron Beams
7. AUTMO«C«J
F. R. Buskirk and J. R. Neighbours
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1*. SUPPLEMENTARY NOTES
It- KEY WORDS (Contlnum on tmvmtmm mldm II n«C(li«ry «j Idmnllty oy block numbmr)
Cerenkov Radiation
Microwave Radiation
Bunched Electrons
20. ABSTRACT (Cominum on tmvmtmm mldm It nmemmsmry mnd Idmnllty 6r mloe* m~a>o«<i
Cerenkov radiation is calculated for electron beams which exceed
the velocity of radiation in a non dispersive dielectric medium.
The electron beam is assumed to be bunched as emitted from a
travelling wave accelerator, and the emission region is assumed to
be finite. Predictions include (a) emission at harmonics of the
bunch rate, (b) coherence of radiation at low frequencies (c) smear-
ing of the emission angle for finite emission regions, (d) explicit
evaluation of cower sroectrum in terms of bunch dimensions. The
do ,;
FORM
AN 7)
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UNCLASSIFIED
CfaCURITV CLASSIFICATION OF THIS PAGECWi^n Dmtm Entmrmd)
results apply to microwave emission from fast electrons in
air or other dielectrics.
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TABLE OF CONTENTS
Page
Introduction 1
Calculation of Poynting Vector 3
Fourier Coefficients of the Current 6
Vector Potential 8
Radiated Fower 11
Cerenkov Angle 13
Discussion of Results 15
a. Effect of Pulse Size 15
b. Shearing of the Cerenkov Angle 16
c. Behavior at high Frequencies Related 16
to Pulse Parameters.
Concluding Remarks; 18
References 20
APPENDICES
A. Derivation of Cerenkov Radiation Al
for a Single Pulse of Charge
B. Derivation of Equation 7 Bl
C. Temporal Structure of the Electron CI
Pulse from a Travelling Wave
Linear Accelerator
D. Form Factors Dl
Distribution List
I NTRODUCTION
The radiation produced by gamma rays incident on ordinary
dielectric materials such as glass was first discovered by
Cerenkov 1 in 1934 and was described in terms of a charged
particle (electron) moving faster than light in the medium by
Frank and Tamm 2 in 1937. A summary of work to 1958 is contained
in the treatise by Jelly-*. An important application is the
Cerenkov particle detector which is familiar in any particle
physics laboratory, and an early and crucial application occurred
in the discovery of the antiproton 4 - .
Because the distribution of intensity of Cerenkov radiation
is proportional to the frequency, the radiation at microwave
frequencies would be low unless beams are intense and bunched so
that coherent radiation by many electrons contributes . Danos^
in 1955 calculated radiation produced by a planar beam passing
above a dielectric interface and a hollow cylindrical beam passing
through a hole in a dielectric. Experimental and theoretical
investigations at microwave frequencies were reviewed by
Lashinsky^ in 1961.
This investigation was motivated by a recent renewed interest
which has included the study of stimulated Cerenkov radiation, in
which the electron may be in a medium consisting of a gas' or a
hollow dielectric resonator^* 9. Recent developments of
electron accelerators for applications such as free electron
lasers (FEL) have aimed toward high peak currents in bunches in
contrast to nuclear and particle physics applications, where low
peak but high average currents are desirable to avoid saturating
1
detectors. The high peak currents in the new accelerators should
yield enhanced Cerenkov radiation, as is calculated in this
paper .
CALCULATION OF THE POYNTING VECTOR
In the following derivation, we consider the Cerenkov
radiation produced in a dispersionless medium such as cases or
other dielectrics, by a series of pulses of electrons such as are
produced by a traveling wave electron accelerator (Linac). The
pulses or bunches are periodic, the total emission region is
finite and the bunches have a finite size.
In determining the radiated power, the procedure is to
calculate the Poynting vector from fields which are in turn
obtained from solutions of the wave equations for the potentials.
Since the current and charge densities entering into the wave
equations are expressed in fourier form the resulting fields and
radiated power also have fourier components. In the derivation, r
is the coordinate at. which the fields will be calculated, r i.s
the coordinate of an element of the charge which produces the
fields and n is a unit vector in the direction of r. We assume
that E(r,t) and B(r,t,) have been expanded in a fourier series;,
appropriate for the case where the source current 1.3 periodic.
Then we have
00
E (tt) -£ e" iaJt 1 tf,a»)
oj=-°° (1)
and a corresponding expansion for B, where w is a discrete
_& — ^
variable and E and B are fourier series coefficients. The
poynting vector S is given by
S = -~E x~B
(2)
and it is easy to show that the average of S in a direction given
A
by a normal vector n i:
T oo
i r n.s at - i y. -lu.
-J /i\=— oo
oj ) x B(r,-(D)
0)=-»
(3)
where T is an integer multiple of the period of the periodic
current .
Letting c = (ye) "1/2 be the velocity of light, in the
medium, the wave equations for A,4> and their solutions are,
2
3 -*■,-*> -*■ »
V — 9 — \ A(r,t) = yj(r,t)
c z 3t z i
I c 2 dt 2 i
'?ir,t) = 1/e p(r,t)
m
3 ,
A(r,t) =y D(r-r', t-t ') J(f',t ') dr 'dt '
«f> (r,t) = 1
e
D(r-?',t-t , )p(r' ,t')d 3 r'dt'
(4)
(5)
where the Green's function D is given by
D(r,t) = 1_ 5 (t-r/c)
4irr
(6)
The vector potential A(r, t) also can be developed in a fourier
series expansion of a form similar to (1) with an expression for
the fourier series coefficients given by
T
A(r,a>) = 1 r dtA(?,t)e 1 '
1 f dtA(r,t)
T I
■ill
d 3 r> j(r' r ai)_l 1__ e iaJ l r - r ' ' /c
r-r'
(7)
Now if we assume that the observer is far from the source so
that jr j>> |r' | for regions where the integrand in (7) is important
we can let |r-r | = *r - n • r in the exponential and jr - r' = :
in the |r - r 1 |~1 factor in (7), obtaining (where n = r/r)
A(r,o>) = Ji_ e^ r/c r[fa 3 r'j;(r' ,a>)e- iiai/crn - t '
47rr JJJ (8)
The fourier series coefficients of the fields are obtained
from those for the vector potential (8) through the usual
relations B = V x A and E = -V<|> - 9A . Under the conditions
3t
leading to (3) the field fourier coefficients are-^:
-* - ^ -l ^
B(r,uj) = ico n x A ( r , oj )
c (9)
-* -i -A -* -k
E(r,co) = -c n x B(r,oj)
(10)
The poynting vector can now be found by using (9) and (10)
in expansions like (1) and then substituting in (2). However it
is more convenient to deal with the frequency components of the
radiated power by substituting (9) and (10) into the expression of
the average radiated power (3).
T 00
1 [ n-Sdt = 1 )
T J y ^—
oj n x A ( r , oo )
c (11)
F OURIER COMPONENTS OF THE CURRENT
The expression (7) for the fourier components of the vector
potential contains the fourier components of the current density.
Consequently it is necessary to examine the form of the current
and its fourier development. Assume the current is in the z
direction and periodic. If the electrons move with velocity v,
and we ignore for the moment the x and y variables, the charge or
current functions should have the general form
f(z,t) = £ e lk z Z £ e ia)t f(k z ,a») (12)
(13)
Z CO
Under the condition of rigid motion,
f (z,t) = f (z-vt)
it is easy to show that
,(k ,cj) = 6 . f (k ) , %
z co,kv~ z (14)
where
fvkj = e 1K z z f (z)dz
*v o Z —
z
(15)
Thus the restrictions of equation (13) reduce the two
dimensional fourier series of eq. (12) to essentially a one
dimensional series (14).
With (14) in mind, the current density associated with the
electron beam from a linear accelerator should be periodic in both
z, t, with a fourier series expansion, but the x and y dependence
should be represented by a fourier integral form:
4«o
J_(r,t) ~ vp(r,t) =
(2 TT)
— CO
dk dk
x y
f ,, \~ i (k-r - cat)
J dk y ^_ e Po
k =- c
z
(k)
16)
where the fourier components of the charge density are
P,o (k) = dx dy - | dze
U 7 )
p (r) is p(r,t) evaluated at t = o and J is assumed to be in
the z direction. Note in eq. (16) that k z and ^ are both
discrete and from (14), oj = k z v.
VECTOR POTENTIAL
The results of the previous section can be applied to the
evaluation of the vector potential and in turn to the fields.
Let the infinite periodic pulse train be made finite,
extending from z = -Z ' to z = +Z ' and let 9 be the angle between n
and A. Then the cross product in (11) can be written
I A *t #"*" \ I a V ioor/c
n x A(r,oj) = smfl -7 — e
CO GO 7 '
f dx' / dy' /" dz' e " ifi ^ ,a)/c
Jm J>™ =^>7 I
JSOO .= 00 — 2
k =- c
t-At2 dk / dk ) v0 o (£)<$, e 1K ' r
(2n) z I x J ot *y k ^ = _^ - k z v /W (18)
But
Z* j. *
dx' J dy' / dz' e ir '- (k ™ /c)
-Z'
(2tt) 2 5(k x - n x aj/c) 5 (k - nu/c)I(Z') ( 19 )
where
Z'
T ,„ n / , , i(k - n co/c)z' •>
I(Z') = / dz' e z z 2 sinGZ . (20)
J-Z' G
and G = k z - n 2 go /c = 00 /v - n z co /c
And thus the cross product term is
/\ •*■ ^ 1 Tj icor/c
n x A(r,a>) I = sinG-^r e v£o(n x Vc, n oi/c, o)/v)I(Z')
(21)
Note that. o> is a discrete variable but from 19, the
continuous variables k x and k y become evaluated at discrete
points .
Returning to (17), a more symmetric form may be obtained by
assuming rhat p (r)# which is periodic in z with period Z, is
actually zero between the pulses. Denoting by p ' ( r) the charge
density of a single pulse, which is zero for z < o and z > Z the
integral on z can be written
z z
, -ik z ,-». f , -ik z ',■»> -, -ik z ' ,a
dz e z p (r; = dz e z p (r) = dz e z p (r)
J (22)
Then (17) the fourier coefficient-, of the charge density, becomes
oo
p (k) =| |fj r d 3 r e" ik ' r : '(r) = |pl ik)
(23)
where p n '(k) is the three dimensional fourier transform of the
single pulse decribed by p Q ' (r) .
Substituting these expressions into (21) gives a final simple
result, for the cross product, form:
|n x A(r,<o)l = sine H e lajr/C (v/Z) Pe ' (k)I(z')
47tr - (24)
where
I(Z' ) = | sin GZ'
G
G = oj/v - n co/c (25 )
k = (n oo/cn a)/c,oo/v)
The components of the Cerenkov E and B fields may now be found by
substituting (24) in (9) and (10).
10
RADIATED POWER
The frequency components of the average radiated power are
obtained by substituting (24) into (II). The negative: frequency
terms equal the corresponding positive frequency terms , yielding a
factor of 2 when the summation range is changed. Multiplying by
r^ converts to average power per unit solid angle, dP/dft ,
yielding
T 00
f§=r 2 i I" n-Sdt = r 2 l£V |nx A(?, M )| 2
00
=1
C
W(<o,n)
(26)
where w(a)/n) is defined to be
2
w (a) ,n) = — 2 g- sin 9 (v /Z ) | p ' (kj j I (Z )
(4it)
(27)
W(o),n) is the power per unit solid angle radiated at the frequency
0), which is a harmonic of the basic angular frequency uj of the
periodic pulse train.
To find P , the total power radiated at. the frequency co ,
W is multiplied by dfi and integrated over solid angle. Note that
n z = cos 9, and as 9 varies, G changes according to (25),
11
w
ith dG = - (co/c) dn so that
d^ = dc{> (c/oj) dG
(28)
Noting that the integral over <£ yields 2"^ , we find the result for
the total radiated power at the frequency oj for all angles
G"
p = r_ _
2 2
}J_ 60 v ^
4tt c
2 J
sin 2 6 jPo(k) | 2 I 2 (Z') 5 dG
(29)
G'
12
CERENKQV ANGLE
The remaining integral over G may now be examined. The
sin^ S and p factors may often be slowly varying compared to
the l2(z') factor, the latter being shown in Fig. 1. For large
Z', the peak in l2(z') becomes narrow, and if the integrand may
be neglected outside the physical range G'<G<G",
G
I,
I (Z ')dG = |4(Z ')
/
i, 2 sinGZ
'}
GZ'
'dG = 4tt
(30)
Then, evaluating the sin 9 factor and p Q (k) at the
point corresponding to G = 0, (which is cos 6 = n z = c/v) shows
that. 9 at. the peak of I(Z' ) is the usual Cerenkov angle 9 C . We
thus obtain for large Z'
P = £- covsin 6 |p (k)| 4ttZ'/Z
oj 4lT C l ~° '
(31)
Now let 2Z'/Z = ratio of the interaction length to pulse spacing
N, the number of pulses. Also Z = v2 tt/ o^ or 2tt/z = '^ / v so
that, (in the large Z' limit),
2 ** 2
P oj = 4tt" 000 °o vsin 9 c IP=^ k )i N.
(32)
To compare with usual formulations, (32) is divided by Nv to
obtain the energy loss per unit path length per pulse:
dE = y_
dx 4tt
o)co sin 8 I p (k) ,
, 1 2
(33)
13
If the pulse is in fact, a point charge, the fourier transform
Dq (k) reduces to q, the total charge per pulse and (33) is
very similar to the usual Cerenkov energy loss formula, where for
a single charge q. the radiation is continuous and the factor
oo cj in (33) is replaced by u) d go. In the present case the pulse
o
train is periodic at angular frequency oj q and the radiation is
emitted at the harmonic frequenies denoted by oj .
14
DISCUSSION OF RESULTS
Equation (29) and the approximate evaluation expressed as
(32) form the main results. Some consequences will now be noted.
a. EFFECT OF PULSE SIZE. The spatial distribution of the
charge in the pulse appears in p^lk) , which is the fourier
transform of the charge distribution. The peak of I 2 (2') in
figure 1 occurs at G = or n z = c/v. Thus at the peak, co/v = n z oj/c
i\o that k, the argument of p' (k), is evaluated at (from 25)
s+S
"*■ A
k = noo/c
(34)
We may also define a charge form factor F(k)
o;<k) = q p(ki (3S)
The form factor F(k) is identically one for a point charge,
and for a finite distribution F(k) = 1 for k =o .
Furthermore F(k) must fall off as a function of k near the
origin if all the charge has the same sign. If the pulse were
spherically symmetric, F(k) would behave as elastic electron
scattering form factors defined for nuclear charge
distributions 1-1 . In that case, the mean square radius <r 2 >
of the charge distribution is given by the behavior of F(k) near
the origin.
F(k) ■*■ 1 - <r 2 > k 2 /6 (spherical pulse) (36)
15
b. SMEARING OF THE CERENKOV ANGLE. For a finite region over
which emission is allowed, namely if 2Z ' is finite, the function
I 2 (Z'), appearing in the integral in (29), will have a finite
width. Since the peak height is 4Z ' 2 and the area is 4tts ' ,
(30), we can assign an effective width 2T = area/ height - ^ I z '
or
r=7T/2z' (37)
Thus the radiation is emitted mainly near G = o (which
corresponds to 9 = 9 C ) hut in a range Ag = +T . But from (25),
00 , 00 . . .
AC = — in 7 = — a(cos9) so that there is a range in cos0 over
c ^ c
wr. ich emission occurs:
a / Q^ C 7T (38)
A (cos9) = - ^—r
00 ZZj
Note that the finite angular width of the Cerenkov cone angle
in (38) has the factor l/co , indicating that the higher harmonics
are emitted in a sharper cone.
c. 3EHAVI0R AT HIGH FREQUENCIES RELATED TO PULSE PARAMETERS.
To be specific let the charge distribution for a single pulse be
given by gaussian functions
1 .-* 2 2 ? 2 9
p (r) = A exp(-x/a - y /a - z7b )
Then F(k) may be found
39)
F(k) = exp(-k 2 a 2 /4 -k 2 a 2 /4 -k 2 b 2 /4)
y Z (40)
Beam pulse parameters could then be determined by measuring the
Cerenkov radiation. For example, fast electrons from an
accelerator in air will e ait with a 9 C of several degrees
in which case k x and k y in (40) can be neglected, giving
F(k) = exp(-k z 2 b 2 /4) = exp[-co 2 b 2 /(4v 2 ) ] (41)
The expected behavio:: of P as a function of go is shown
00
qualitatively in Fig. 2 as a linear rise at. low frequencies
followed by a fall off at. higher frequencies, the peak occurring
at
(o = v/b
m
(42)
Furthermore, a different behavior would be expected at. very
high frequencies. The formulation from the beginning represents
coherent, radiation from all charges, not only from one pulse, but.
coherence from pulse to pulse. F(k) then describes interference
of radiation emitted from different, parts of the pulse, but note
that expressions (29} and (32) will still be proportional to
q2 - n 2 e 2 w here n ± s the number of electrons in a pulse.
Thus the n^ dependence of P indicates coherence. But. above
CO
some high frequency go j_ such that oj . /c - 2\\ I % , where I is the
mean spacing of electrons in the cloud, the radition should switch
over to incoherent radiation from each charge and P should be
00
proportional to n. The incoherent radiation should then rise
again as a function of au .
17
CONCLUDING REMARKS
The general results presented here describe the Cerenkov
radiation produced by fast electrons produced by a linear
accelerator. For an S band Linac operating at about 3Ghz (10 cm
radiation) , the electron bunches are separated by 10 cm and would be
about 1 cm long at 1% energy resolution. Microwave Cerenkov
radiation is expected and has been seen in measurements at the Naval
Postgraduate School Linac.
Two types of measurements were made. In measurements of Series
A, an X-band antenna mounted near the beam path, oriented to
intercept the Cerenkov cone, was connected to a spectrum analyser.
Harmonics 3 through 7 of the 2.85 GHz bunch frequency were seen but
power levels could not be measured quantitatively. Harmonics 1 and 2
were below the wave guide cut off. In the series B measurements, the
electron beam emerged from the end window of the accelerator, and
passed through a flat metal sheet 90 cm downstream oriented at an
angle <j> from the normal to the beam. The metal sheet defined a
finite length of gas radiator, and reflected the Cerenkov cone of
radiation toward the accelerator but rotated by an angle 2<t> from the
beam line. A microwave X-band antenna and crystal detector with
response from 7 to above 12 GHz could be moved across the (reflected)
Cerenkov cone as a probe.
As mentioned earlier, the series A measurements showed the
radiation is produced at the bunch repetition rate and its harmonics.
Series B measurements performed with several antennas always
indicated a broadened Cerenkov cone with strong radiation occuring at
angles up to 10°, well beyond the predicted Cerenkov angle of 1.3°.
18
Since a broad band detector was used it was impossible to verify
the prediction (see eq . 38) that the broadening cf the cone should
depend on the harmonic number. However, it should be noted that
incoherent radiation by a beam of lu A at 9 c = 1.3° for a 1
meter path in air would be about 10~^ watts at microwave
frequencies so that observation of any signal by either method A
or B clearly demonstrated coherent radiation by the electron
bunches .
Many of the concepts were clearly notea by Jelly in his
treatise (Jelly^, Section 3.4 especially). The form factor was
noted but a general expression was not given. In fact, the form
factor quoted by Jelly represents the special case of a uniform
line charge of length L' with a projected length L=L'cos9 c in
the direction of the radiation. The coherence of the radiation
from the bunch was noted but no broadening of the: cone nor
harmonic structure were developed.
Casey, Yen and Kaprielian-^ considered an apparently
related problem in Cerenkov radiation, in whj ch a single electron
passes through a dielectric medium, where a spatially periodic
term is added to the dielectric constant. The result is radiation
occurring even for electrons which do not exceed the velocity of
light in the medium, and at angles other than the Cerenkov cone
angle. The non-Cerenkov part of the radiation is attributed to
transition radiation.
In the present paper, the transition radiation associated
with the gas cell boundaries is included, and radiation appears
outside the Cerenkov cone.
19
If the electron velocity were lower so that v/c were close to but
less than unity, the peak in I would be pushed to the left in Fig.
1, such that cos 9 c = v/c would be larger that 1. But. the tails of
the diffraction function I would extend into the physical range 1 _<
cos 8 <_ - 1 , and this would be called transition radiation and he
ascribed to the passage of the electrons through the boundaries of
the gas cell. Now return to the case v/c > i, with the situation as
shown in Fig. 1. The radiation is then a combination of Cerenkov and
transition radiation. The formalism of Reference 12 does admit a
decomposition into the two types of radiation, but is inherently much
more cumbersome.
As a final remark, one might extend the analysis further in the
region near w ^ . Consider electron bunches emitted frcm a
travelling wave Linac, which could be 1 cm long spaced 10 cm apart.
Let these bunches enter the wiggler magnet of a free electron laser
(FEL) . Then, if gain occurs, the 1 cm bunches would be subdivided
into bunches of a finer scale, with the spatial r-cale appropriate to
the output wavelength of the FEL. 13 If the (partially) bunched
beam from the FEL were passed into a gas Cerenkov cell, then the
observed radiation should be reinforced because cf partial coherence,
at the FEL bunch frequency and harmonics. This would lead to bumps
in the spectrum in the region near oj ^ .
20
ACKNOWLEDGEMENTS
This work was made possible by the Office of Naval Research
W. B. Zeleny provided much insight into the theory for point
charges, and the experiments performed by Ahmet Saglern, Lt . ,
Turkish Navy, provide preliminary substantiation of the work.
21
REFERENCES
1. Cerenkov, P., Dcklady Akad-Nauk, S.S.S.R., 2 451, (1934).
2. Frank, I.M. and Tamm, I., Doklady Akad-Nauk 3.S.S.R., L4 109,
(1937) .
3. Jelly, J. V., "Cerenkov Radiation and its Applications".
Pergammon Press, London, (1958).
4. Chamberlain, 0., E. Segre, C. Wiegand and T. Ypsilantis, Phys .
Rev. , 100 947, (1955) .
5. E'anos, M.J., J. Appl. Phys, 26_ 1 , (19 55).
6. Laskinsky H. "Cerenkov Radiation at Microwave Frequencies"
in advances in Electronics and Electron Physics, L. Marton,
(ed) Academic Press, New York, Vol. XIV, pp. 265, (1961).
7. Piestrup, M.A., R.A. Powell, G.B. Rothbart, C.K. Chen and R.H.
Fantell, Appl. Phys. Lett., 28. 92, (1976).
S. Vialsh, J.S., T.C. Marshall and 3. P. Schlesinger, Phvs . Fluids,
20 709 1977.
9. Valsh, J.E. "Cerenkov and Cerenkov-Raman Radiation Sources" in
P hysics of Quantum Electronics , S. Jacobs, M. Sargent and
M. Scully (ed) Vol 7, Addison Wesley, r eading, Mass., (1979).
10. Jackson, J.D., "Classical Electrodynamics", (2 nc ^ Edition)
John Wiley, New York 1975 (pp.4) Sec. 9.2 pp. 394ff.
11. Hofstadter , R. , Ann. Rev. Nuc Sci, Vol 7. pp. 2 31, (1937).
12. Casey, K.F., C. Yeh and Z.A. Keprielian, Phvs. Rev. 140 B 768
(1965) .
13. Hopf, F.A., P. Meystre and M.O. Scully, Phys. Rev. Lett 3_7
1215 (1976) .
22
r=TT/2Z
J!
*G
* e
Tf
^i
igure 1.
Qualitative Behavior of the Function I 2 (Z'). Both
the function G, from Eq. 25 in the text, and the
emission angle are displayed as independent
variables. G' and G" are upper and lower limits.
23
Pu
» OJ
Fiqure 2. Schematic Behavior of Power Emitted as a function of
Angular Frequency
24
F 3 Eo
o -
— r
♦2 AY
W
o
i — i —
2TT
-5- ^;
Figure 3. Structure of charge pulse from a travelling wave
accelerator. y is the phase angle of an electron
relative to the peak of the travelling wave
accelerating field. Electrons in the range +A^are
passed by magnetic deflection system.
25
APPENDIX A
DERIVATION OF CERENKOV RADIATION FOR A SINGLE PULSE OF
CHARGE .
Let the pulse be described by
p' (r,t) = p ' Q (r- vt) Al
Both k z and go are continuous variables in this case; "v is again
along the z axis. If we expand in terras of a four dimensional
fourier integral,
p'(r,t) = l/(2ir) 4 e 1 ^ " k - r ) o ' (k, w) d 3 kdoj A2
It may be shown that the condition Ai gives:
p'(k,aj) = 2tt6(o) - k z v) p' Q (k) A3
where p' (k) is the three dimensional spatial tranform of p 1
evaluated at t = o. All the fields have fourier integral rather
than fouries series expansions and the energy radiated per unit
solid angle become
2 ^ ~ £ X ] - -, u f 2,
dt n • S = t" ,,,, 2 — (o dw
2tt (4tt) c J _
'Jd 3 r' J df e i(ct'-n.?')o)/c. x ?( *, tl) |
A4
m |2
W ("> , n ) do)
Al
The integrand is a symmetric function of co so that
W
<-'*-^3*"\ffl) r * w
icu(t' -n-r' /c) ^ £,«»■, . , ,
e nxJ(r ! ,t')
A5
where
1 y ,* -, 2.,2
= — -, — co vn x v) M
, - 3 c
lOTT
M =
,3 , , Mcot'-n-r /c) ,,-"", ,, \
d r d-c e p (r , t )
A6
Now we may write p'(r',t') in a fourier integral representation
•(r\t')
(2ir)
J.
*3. , , , ,,.-*, M -i(o) 't'-k' -r ')
a k dco p ( .< , co ) e
A7
Inserting eq . A3 into eq. A7 and the result into eq A6, the
integral over d^ ' involves only exponentials and yields
(2rr)3 o3(y v ' _ om/c) , so that eq A6 becomes
II d 3 k'dco'e i(a) " a) " )t, 6 3 (k' - ncu/c:
5 (aj'-k. 'v)p' (k')
Now the integral over go' may be done; because of the 6 function,
to' is evaluated at k z 'v.
M =
/dt^AV
i i
L(co-k v) t £.3.^, a. . .,,-*..
z (S (k ' - con/c) p ' (k ' )
-v o
A2
Now do the integrals over k x ' , k y ' and k z ', noting that
k ' appears in the exponential, but k ' and k ' do not,
u _ [,, icot' -iwt'n
M = / dt e e :
v / c ' / / / / s
; p (am /c,am /C/um /c)
~ x y z
This may be written as
M = dt e p (oon/c) A8
where
H = i-n v/c
z A9
If we let the time interval be finite, from -T to + T, the
integral is easily done:
2_
iM = — sin ojHTPo (nco/c) A10
M = 4T sin cjHt I Po (noo/c) | A ]_l
(coHT) 2
A3
This result, eq. All may be inserted into A5 for oo • The
factor n x v is just sin 9 where 9 is the angle between the
radiation and the beam axis.
W(oj, n ) = —- — -H- (a sin Z 9 4T Z sin coHT' l p ' (nca/c) | Al3
16 * C (wHT) 2 '
W is the energy radiated per unit solid angle per unit
angular frequency, co • To proceed to the total energy, multiply
by d ft (solid angle) and integrate. But n z = cos 9 so that dQ
may be related to cH :
d ft = d(cos 8)dft= - — dHdft A , ,,
v A14
Tii'i functions in eq . A13 do not contain <j> so that integration
over G yields 2 it. Thus:
wi , j n 1 y 2 m 2f . 2 , ',2 sin coHT ,„
A(co,nidft = — co T /sin 9|p | dH 5
2rr 2 ~ (ooHT) 2
The sm 2 goHT / (coHT) 2 factor in the integral is peaked at H = 0,
c
which by eq. A9 is at n z = cos 9 = — , or the usual Cerenkov
angle, 9 c . This function is more strongly peaked about H =
for large values of T, and in fact, for large T we may evaluate
sin^ and P ' at the point corresponding to H =0. Then the
integral
oo
J
2 2
dx sin (ax) /(ax) = rr/a
A4
may be used to evaluate eq . A15, yielding
ffwdfi = i- -| 2Tsin 2 9c|p;(Vc)| 2
JJ
A16
The emmission was assumed to occur in a time interval from -T to
•KT ; accordingly dividing by 2T yields a rate of emmission, and
multiplying by v converts to emmission per unit path length. Thus
we obtain, for the large T limit:
dxdco
GO) =
— ujdco sin 8 |o'(nco/c)|
A17
where d^E/^^ is the energy emitted per unit path length
per unit angular frequency range co .
The corresponding expression for T not. large is
H"
d 2 E , U , /coT
-= — -=— dco = -: — go dco —
dxdco 4tt it
sin 9jp * (noj/c) J sin^coHT
H' (coHT)
Aia
where H" and H' are the value of H corresponding to 9=0 and
5 = tt respectively.
Equations Al 7 and A18 then describe the energy radiated per
unit path length and per unit angular frequency range. For the
non periodic (single) pulse the radiation has a continuous
freauency spectrum. For a point charae q, p' (k) is identically q
and the usual Cerenkov formula is obtained. Equation A17 is
quoted by Jelly, but only with the form factor corresponding to a
uniform line charge of length L' .
APPENDIX B
DERIVATION OF EQUATION 7
Equation 7 is derived for the case in which J(r,t) is
— »
expanded in fourier series. Let the fourier coefficient for A
be given by:
T
r
A(r\o)) = i J dt A(r\t) e icut B1
o
Assume that the green's function solution for A(r,t) is given:
(r,t) = uj r /Td 3 r' |
dt' J(r',t) D(r - r', t - t)
32
where
D(r,t) - ~ Sit - r/c)
Let the current density be expanded in a fourier series
B3
J(?\f ) =Z e- iw ' t, J(?',a J ) B4
co'
Then insert 32, B3 and B4 into 31 to obtain
t (*.«> = H. J dt e 1 ^ JjfdV Jc.f ^ ,^— p-
B5
e ia),t ' J(r« ,03')
_>
z
Bl
Do
-ioi 't'
Jl dt ' , note that t' appears in the <5 function and in
The result is t' is evaluated at t' = t - j r-r 1 l/c.
T
u
A(r,co) = —
~ T
dt e
loot
d J r'
1 1
4tt i-» •»'
r-r
B6
I
0)
-ico 't ico ' j r-r ' I /c ^, -^, 1,
e e ' ' J(r ,a> )
Do the integral on t, note that
T
kj dt •"
L (co-co ') L
= 6
CO CO
B7
Then do the sum on co '
A(r ,co; =
4tt
d r
J(r ,co; e
ico I r-r 1 /c
r-r
38
This proves the desired result, B8 is equation 7 as used in
the main text.
Appendix C
TEMPORAL STRUCTURE OF THE ELECTRON PULSE FROM A TRAVELLING
WAVE ACCELERATOR.
Assume that the energy of a single electron emerging from a
linac with phase $ relative to the travelling wave field is
CI
E = E cos-J;
o
This relation is shown on fig. 3, along with some dots
representing electrons rear the maximum energy E c , with phases
clustered about 'b = o and ty - 2^. Two bunches, separated by a
phase difference of 2t, are separated by a time T-i = 1/f where
f is the accelerator frequency, which is f Q = 2.85 x 10^ Hz
for a typical S-band accelerator of the Stanford type.
If a deflection system with energy resolution slit passes
only energies E from E Q to E Q - E the corresponding range of phase
Aip is
C2
AE = E - E = E (1- cosAijj)
o c
For AiJj small, this reduces to
AE = (Ai|;) 2 c3
E o " 2
CI
The temporal pulse length T2 is
T n = 2A<|; T,/2
C4
2 x
or
T_ = T • 2u^/2tt
If C3 is used to evaluate Aii> in terms of the fractional
energy resolution AE/E Q
O 7T CD
For 1% energy resolution, 1 , 2 /T 1 is about 1/20. The
electrons thus emerge in short bunches, and the charge and
current, when expressed in a fourier expansion, should have very
strong harmonic content up to and aoove the 20th harmonic.
C2.
Appendix D
FORM FACTORS
This section provides details and examples of form factors
for various charge distributions. From the main text, F differs
only from P, the fourier transform of P, by the total charge q of
the bunch, so that for k = o, F reduces to unity. Thus we define
£ /// d3r P(r)e i ^'"
F(k) = £ / M d J r Pirje^ 1 L
q
For spherically symmetric charge distributions , let k*r = kru,
where u is the cosine of the angle between k and r. In spherical
coordinates, d^r = dudyr 2 dr. Then we find,
CO
F ( k ) =1±I / dr r p ( r )
r~< 1, -J
sin Kr D2
q k
For k very small, sin x may be replaced by x - x 3 /6 and we
have
*°
F(k) = 1 ±1 / dr r p(r)[kr - k 3 r 3 /6] D 3
q k J
Then the two terms in the square bracket lead to separate
integrals, the first term being unity and the second is similar to
the integral used to calculate the mean square radius, <r~>,
exceot for a factor k 2 /6. Thus we have
F(k) = 1 - k 2 <r 2 >/6
D\
D4
For a uniform spherical charge distribution of radius R, as
well a a spherical shell of radius k, the integral D2 may be
performed easily
F(k) = 3 (sin kR - kR cos kR)
(Solid sphere)
(kR) T
D5
F(k) = I sin(kR) (Spherical shell)
kR
D6
For a line charge concentrated on the z axis, we may return
to Dl and let p(r) = 5(x) <5(y) p"(z), so that
F(k) = 1 dz p"(z) e ikz (line charge)
D7
Q
F(k) - j sin ( _) (uniform line charge
kZ 2
of length Z)
D8
Distorted spherical symmetry may be said to occur if the
scale transformation z' = pz serves to make p spherically
symmetric in tine prime system. Let F g be the form factor
calculated by 2 in the prime frame. It is simple to show that
F(k x , k y , k z ) = F s (k^, k^, k^/p) D9
DZ
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